Improvement of effective Erdos-Wintner theorem for Zeckendorf expansions
Johann Verwee
TL;DR
This work addresses the problem of establishing an effective Erdos--Wintner theorem for Zeckendorf expansions under weaker tail assumptions. The authors develop a paired-block transfer-matrix method that extracts a common linear phase along the dominant direction and analyzes the product to second order, yielding quadratic-tail or split-tail bounds that require only $\sum_j f(F_j)^2<\infty$ rather than absolute convergence of $\sum_j f(F_j)$. The main contributions are the precise $(1,1)$-entry formula, the cancellation of the linear term, and the resulting tail bounds that feed into a Berry--Esseen smoothing framework, along with a concrete example showing the theory applies when $\sum_j |f(F_j)|$ diverges. Taken together, these results extend the applicability of effective Erdos--Wintner type bounds to a broader class of Zeckendorf-additive functions and provide robust quantitative control via a quadratic or split tail, with implications for the convergence rate to the limiting distribution $F$ and its concentration function $Q_F$.
Abstract
We revisit the effective Erdos-Wintner theorem for Zeckendorf expansions. Drmota and the author obtained a uniform Kolmogorov bound whose error involves $T\sum_{j>L-2h}|f(F_j)|$, which assumes absolute convergence of the linear tail $\sum_j f(F_j)$. We remove this assumption. Grouping the transfer matrices in pairs and working to second order on the logarithm of the product, after extracting the common linear phase along the dominant direction, yields a quadratic tail $T^2\sum_{j>L-2h} f(F_j)^2$, or, in a flexible variant, the split tail $T\sum_{|f(F_j)|>1/T}|f(F_j)| + T^2\sum_{|f(F_j)|\le 1/T} f(F_j)^2$. Either form requires only $\sum f(F_j)^2<\infty$.